A hypergeometric approach, via linear forms involving logarithms, to criteria for irrationality of Euler’s constant

2009/05/16 by Jonathan Sondow, Sergey Alekseevich Zlobin · 1 citation
Mathematics · #Advanced Mathematical Identities #Mathematical functions and polynomials #Analytic Number Theory Research #Mathematics #Hypergeometric function #Irrationality #Constant (computer programming) #Logarithm #Euler's formula #Pure mathematics #Hypergeometric distribution #Hypergeometric identity #Generalized hypergeometric function #Type (biology) #Series (stratigraphy) #Bilateral hypergeometric series #Mathematical analysis #Algebra over a field #Hypergeometric function of a matrix argument #Law

paper · pdf · doi:10.2478/s12175-009-0127-2

openalex publication_date 2009/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Abstract Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.

Citations

• Criteria for irrationality of Euler’s constant

Cited by

• Euler's constant [wikipedia]

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